We can now describe the direction and strength of trends in bivariate data, but there is a more precise way of measuring how strong a linear model fits the data. It is called the correlation coefficient (also called Pearson's correlation coefficient), and it is represented by the letter $r$r. It measures how close bivariate data is to a straight line, and also tells us whether the line has a positive or negative gradient.
Calculating the correlation coefficient by hand is very time-consuming. People usually use technology (either a calculator or a computer) to calculate the coefficient, and then do the more difficult task of interpreting their result.
Direction of a linear relationship
The direction of a linear relationship is given by the sign of the correlation coefficient.
- If $r$r is positive then there is evidence of a positive linear relationship.
- If $r$r is negative then there is evidence of a negative linear relationship.
Positive linear relationship
$r>0$r>0
Negative linear relationship
$r<0$r<0
Strength of a linear relationship
The strength of a linear relationship is given by the value of $r$r. The closer $r$r is to $1$1 o $-1$−1, the stronger the relationship is likely to be.
- No linear
- relationship
- $$
- Weak linear
- relationship
- $$
- Moderate linear
- relationship
- $$
- Strong linear
- relationship
- $$
There are some special cases to consider.
- If $r=1$r=1 then all of the data points lie on a line with positive gradient.
- If $r=-1$r=−1 then all of the data points lie on a line with negative gradient.
- If $r$r is close to $0$0 then there is probably no linear relationship. This could mean that the variables are unrelated, but it could also mean that there is a non-linear relationship between them.
- Occasionally we might find that the correlation coefficient is impossible to calculate. This usually indicates that one or both of the variables is actually constant.
Perfect positive linear
relationship
$r=1$r=1
Perfect negative linear
relationship
$r=-1$r=−1
Non-linear
relationship
$r=0.1$r=0.1
Constant
relationship
$r$r is undefined
Using technology to calculate the correlation coefficient
| Many websites have a correlation coefficient calculator, where you enter the ordered pairs in your data set and the calculator returns the coefficient. The correlation coefficient is often labelled $r$r or $R$R. Image 1 shows pairs of values of $x$x and $y$y entered separately but in corresponding order to calculate $R$R. | Of course you can also use a graphics calculator to enter the ordered pairs, and have it return the correlation coefficient value. Image 2 shows that in the Statistics mode, you can again enter the $x$x and corresponding $y$y-values in lists 1 and 2 and retrieve the correlation coefficient $R$R. |
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The important work lies in interpreting these calculations, and what they mean for the strength of the relationship between the variables.
Practice questions
Question 1
Calculate the correlation coefficient for the bivariate data in the table below to the nearest two decimal places.
| $x$x | $7$7 | $12$12 | $6$6 | $19$19 | $8$8 | $16$16 | $20$20 | $9$9 | $11$11 | $19$19 |
|---|---|---|---|---|---|---|---|---|---|---|
| $y$y | $4$4 | $8$8 | $2$2 | $12$12 | $5$5 | $8$8 | $9$9 | $5$5 | $4$4 | $10$10 |
Question 2
Describe the relationship between variables with a correlation coefficient of $-0.34$−0.34.
No linear relationship
AStrong negative linear relationship
BWeak positive linear relationship
CStrong positive linear relationship
DWeak negative linear relationship
E
Question 3
The linear relationship between a set of data for variables $x$x and $y$y has a correlation coefficient of $0.3$0.3.
The linear relationship between a set of data for variables $x$x and $t$t has a correlation coefficient of $-0.9$−0.9.
Do the two linear relationships have the same direction?
Yes
ANo
BWhich relationship has a stronger correlation?
Between $x$x and $y$y.
ABetween $x$x and $t$t.
B